Smooth deformation of rigged vertex group
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What's going on here is that the vertices you have painted only have weights that move them, and none trying to hold them in place. When weight painting, you're essentially telling blender how much a certain bone will affect a vertex. Think of it like a tug-of-war. If there's only one person pulling, it doesn't matter how strong they are, they're going to win.

Thus, to solve this problem, you'll need to add a bone (say a head bone) that the stationary parts of your model are assigned to. Then it's just a matter of blending the two weights. A trick for this is to move the bones to a point where it doesn't deform well, then paint it until it does. Then you can reset the bone (Alt+G and Alt+R). You may find the blur brush handy as well, as that can eliminate the sharp transitions between colors that you see in the third image.